MA5251: Spectral Methods & Applications

1. MA5251: Spectral Methods & Applications Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao

2. Contents • Introduction and Preliminaries • Some numerical examples • Review of different numerical methods of PDE • Historical background of spectral methods • Some examples of spectral methods • Fourier series and orthogonal polynomials • Review of iterative solvers and preconditioning • Review of time discretization methods • Spectral-Collocation Methods • Introduction • Differentiation matrices • Fourier, Chebyshev collocation methods

3. Contents • Spectral-Galerkin methods • Introduction • Fourier spectral method • Legendre spectral method • Chebyshev spectral method • Error estimates • Spectral methods in unbounded domains • Introduction • Hermite spectral method • Laguerre specreal methods,….

4. Contents • Applications • In fluid dynamics • In heat transfer • In material sciences • In quantum physics and nonlinear optics • In plasma and particle physics • In biology • …………..

5. Dynamics of soliton in quantum physics

6. Wave interaction in plasma physics

7. Wave interaction in plasma physics

8. Wave interaction in plasma physics

9. Wave interaction of plasma physics

10. Wave interaction in particle physics

11. Vortex-pair dynamics in superfluidity

12. Vortex-dipole dynamics in superfluidity

13. Vortex lattice dynamics in superfluidity

14. Vortex lattice dynamics in superfluidity

15. Vortex lattice dynamics in BEC

16. Main numerical methods for PDEs • Finite difference method (FDM) – MA5233 • Advantages: • Simple and easy to design the scheme • Flexible to deal with the nonlinear problem • Widely used for elliptic, parabolic and hyperbolic equations • Most popular method for simple geometry, …. • Disadvantages: • Not easy to deal with complex geometry • Not easy for complicated boundary conditions • ……..

17. Main numerical methods • Finite element method (FEM) – MA5240 • Advantages: • Flexible to deal with problems with complex geometry and complicated boundary conditions • Keep physical laws in the discretized level • Rigorous mathematical theory for error analysis • Widely used in mechanical structure analysis, computational fluid dynamics (CFD), heat transfer, electromagnetics, … • Disadvantages: • Need more mathematical knowledge to formulate a good and equivalent variational form

18. Main numerical methods • Spectral method – This module • High (spectral) order of accuracy • Usually restricted for problems with regular geometry • Widely used for linear elliptic and parabolic equations on regular geometry • Widely used in quantum physics, quantum chemistry, material sciences, … • Not easy to deal with nonlinear problem • Not easy to deal with hyperbolic problem • …..

19. Main numerical methods • Finite volume method (FVM) – MA5250 • Flexible to deal with problems with complex geometry and complicated boundary conditions • Keep physical laws in the discretized level • Widely used in CFD • Boundary element method (BEM) • Reduce a problem in one less dimension • Restricted to linear elliptic and parabolic equations • Need more mathematical knowledge to find a good and equivalent integral form • Very efficient fast Poisson solver when combined with the fast multipole method (FMM), …..

20. Historical background • Method of weighted residuals (MWR) – Finlayson & Scriven (1966) • Trial functions (or expansion or approximation functions): are used as the basis functions for a truncated series expansion of the solution. • Test functions (or weight functions): are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion. • This is achieved by minimizing the residual, i.e. the error in the differential equation produced by using the truncated expansion instead of the exact solution, with respect to a suitable norm. • An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions.

21. Historical background • Trial functions • Spectral method: infinitely differentiable global functions, i.e. eigenfunctions of singular Sturm-Liouville problems • Finite Element Method (FEM): partition the domain into small elements, and a trial function (usually polynomial) is specified in each element and thus local in character & well suited for handling complex geometries. • Finite Difference Method (FDM): similar as FEM. • Test functions • Spectral methods: three different ways • FEM: similar as trial functions • FDM: Dirac delta functions centered at the grid points

22. Historical background • Different test functions of spectral methods • Galerkin method: same as the trial functions which are infinitely smooth functions & individually satisfy the boundary conditions. The differential equation is enforced by requiring that the integral of the residual times each test function be zero. • Collocation method: Dirac delta functions centered at the collocation points. The differential is required to be satisfied exactly at the collocation points. • Spectral tau method: Similar as the Galerkin method except that no need the trial and test functions satisfy the boundary conditions. A supplementary set of equations is used to apply the boundary conditions.

23. Historical background • Collocation approach (simplest of the MWR) –Slater (1934); Kantorovic (1934); Frazer, Jones and Skan (1937). • Proper choice of trial functions and distribution of collocation points – Lanczos (1938) • Orthogonal collocation method – Clenshaw (1957); Clenshaw and Norton (1963); Wright (1964). • Earliest application of spectral methods to PDE – Kreiss and Oliger (1972)—Fourier method; Orszag (1972) –pseudospectral. • Spectral-Galerkin method – Silberman (1954) in meteorological modeling; Orszag (1969, 1970); etc.

24. Historical background • Theory of spectral method -- Gottlieb and Orszag (1977) • Symposium Proceedings – Voigt, Gottlieb and Hussaini (1984) • First International Conference on Spectral and High Order Methods (ICOSAHOM) -- Como, Italy in 1989. It becomes series conference every three years. The next one is http://www.math.ntnu.no/icosahom/